Let L be a self-adjoint invertible operator in Hilbert space such that − 1 is p -summable. Under certain discrete dimension spectrum assumption on , we study the relation between (regularized) Fredholm determinant, det ( I + z ⋅ ) one hand and zeta regularized ζ other. One of main results formula = exp ⁡ ∑ j ! d log | 0 . We show derivatives can expressed terms values heat trace coefficients Fu...

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H = AS + AS + B (with S the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E−, E+], E− < E+, we prove that A and B are certa...

Diagonalization is one of the most important topics one learns in an elementary linear algebra course. Unfortunately, it only works on finite dimensional vector spaces, where linear operators can be represented by finite matrices. Later, one encounters infinite dimensional vector spaces (spaces of sequences, for example), where linear operators can be thought of as ”infinite matrices”. Extendin...

Abstract. We consider functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B ∞,1 (R), then we have the following Lipschitz type estimate in the trace norm: ‖f(A1, B1)− f(A2, B2)‖S1 ≤ const(‖A1 −A2‖S1 + ‖B1 −B2‖S1). However, the condition f ∈ B ∞,1 (R) does not imply the Lipschitz ...

These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensio...

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := −[pu∇]∆ + qu, on an arbitrary, bounded time-scale T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L(Tκ), in such a way that the resulting operator is self-adjoint,...

For self-adjoint operators A,B, a bounded operator J , and a function f : R → C, we obtain bounds in quasi-normed ideals of compact operators for the difference f(A)J − Jf(B) in terms of the operator AJ − JB. The focus is on functions f that are smooth everywhere except for finitely many points. A typical example is the function f(t) = |t|γ with γ ∈ (0, 1). The obtained results are applied to d...

We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions families such operators. In one direction, we show that any homogeneous variational scalar Riemannian invariant (CVI) induces these another use ambient metric to alternative certain produced this way, which is a self-adjoint, fourth-order, tridifferential operator should...

This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are essentials in Quantum Mechanics: unbounded operators, multiplication operators, self-adjointness, spectrum, functional calculus, spectral measures and von Neumann’s Spectral Theorem. 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . ...